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OPEN
Received: 17 February 2017 Accepted: 5 May 2017 Published: xx xx xxxx
Quantifying non-ergodicity of anomalous diffusion with higher order moments Maria Schwarzl1, Aljaž Godec1,2 & Ralf Metzler 1 Anomalous diffusion is being discovered in a fast growing number of systems. The exact nature of this anomalous diffusion provides important information on the physical laws governing the studied system. One of the central properties analysed for finite particle motion time series is the intrinsic variability of the apparent diffusivity, typically quantified by the ergodicity breaking parameter EB. Here we demonstrate that frequently EB is insufficient to provide a meaningful measure for the observed variability of the data. Instead, important additional information is provided by the higher order moments entering by the skewness and kurtosis. We analyse these quantities for three popular anomalous diffusion models. In particular, we find that even for the Gaussian fractional Brownian motion a significant skewness in the results of physical measurements occurs and needs to be taken into account. Interestingly, the kurtosis and skewness may also provide sensitive estimates of the anomalous diffusion exponent underlying the data. We also derive a new result for the EB parameter of fractional Brownian motion valid for the whole range of the anomalous diffusion parameter. Our results are important for the analysis of anomalous diffusion but also provide new insights into the theory of anomalous stochastic processes. Superresolution microscopy allows unprecedented insight into the motion of fluorescently labelled, single molecules in complex liquid environments and even inside living biological cells. The observed tracer dynamics also reveals new insights into the physical properties of the systems and thus provides a handle for the modelling of followup processes such as molecular reactions in the system. Concurrently, due to ever increasing computational power, large scale simulations uncover longer and longer time windows of the atomistic or coarse grained dynamics in molecular systems1–6. In complex systems such as living biological cells one often observes systematic deviations of the tracer dynamics from Brownian motion. Thus, anomalous diffusion characterised by the power-law scaling x 2(t ) Kαt α
(1)
of the mean squared displacement (MSD) emerges, where Kα is the generalised diffusion coefficient of dimension cm2/secα. According to the magnitude of the anomalous diffusion exponent α one distinguishes subdiffusion for 0 0 and t1 ≠ t245, 61–63 The anomalous diffusion exponent α used here relates to the Hurst exponent, often encountered in the discussion of FBM, via α = 2H. Due to the sign of the factor (α − 1) subdiffusive FBM for 0
OPEN
Received: 17 February 2017 Accepted: 5 May 2017 Published: xx xx xxxx
Quantifying non-ergodicity of anomalous diffusion with higher order moments Maria Schwarzl1, Aljaž Godec1,2 & Ralf Metzler 1 Anomalous diffusion is being discovered in a fast growing number of systems. The exact nature of this anomalous diffusion provides important information on the physical laws governing the studied system. One of the central properties analysed for finite particle motion time series is the intrinsic variability of the apparent diffusivity, typically quantified by the ergodicity breaking parameter EB. Here we demonstrate that frequently EB is insufficient to provide a meaningful measure for the observed variability of the data. Instead, important additional information is provided by the higher order moments entering by the skewness and kurtosis. We analyse these quantities for three popular anomalous diffusion models. In particular, we find that even for the Gaussian fractional Brownian motion a significant skewness in the results of physical measurements occurs and needs to be taken into account. Interestingly, the kurtosis and skewness may also provide sensitive estimates of the anomalous diffusion exponent underlying the data. We also derive a new result for the EB parameter of fractional Brownian motion valid for the whole range of the anomalous diffusion parameter. Our results are important for the analysis of anomalous diffusion but also provide new insights into the theory of anomalous stochastic processes. Superresolution microscopy allows unprecedented insight into the motion of fluorescently labelled, single molecules in complex liquid environments and even inside living biological cells. The observed tracer dynamics also reveals new insights into the physical properties of the systems and thus provides a handle for the modelling of followup processes such as molecular reactions in the system. Concurrently, due to ever increasing computational power, large scale simulations uncover longer and longer time windows of the atomistic or coarse grained dynamics in molecular systems1–6. In complex systems such as living biological cells one often observes systematic deviations of the tracer dynamics from Brownian motion. Thus, anomalous diffusion characterised by the power-law scaling x 2(t ) Kαt α
(1)
of the mean squared displacement (MSD) emerges, where Kα is the generalised diffusion coefficient of dimension cm2/secα. According to the magnitude of the anomalous diffusion exponent α one distinguishes subdiffusion for 0 0 and t1 ≠ t245, 61–63 The anomalous diffusion exponent α used here relates to the Hurst exponent, often encountered in the discussion of FBM, via α = 2H. Due to the sign of the factor (α − 1) subdiffusive FBM for 0
